Problem 30 - Algebra and Trigonometry

Question

Solve the inequality. Express the answer using interval notation. $$ |5 x-2|<6 $$

Step-by-Step Solution

Answer

The solution is \((-\frac{4}{5}, \frac{8}{5})\).

Step 1: Understand Absolute Inequality

The inequality \(|5x - 2| < 6\) involves an absolute value. This can be understood as the expression inside the absolute value lying within a distance of 6 from 0. Thus, we split it into two separate inequalities: \(5x - 2 < 6\) and \(5x - 2 > -6\).

Step 2: Solve the First Inequality

For the inequality \(5x - 2 < 6\), we first add 2 to both sides to isolate the term with the variable. This gives us \(5x < 8\). Next, divide each side by 5 to solve for \(x\). This results in \(x < \frac{8}{5}\).

Step 3: Solve the Second Inequality

For the inequality \(5x - 2 > -6\), begin by adding 2 to both sides, resulting in \(5x > -4\). Divide each side by 5 to solve for \(x\). This results in \(x > -\frac{4}{5}\).

Step 4: Combine the Results

We now have the two inequalities \(x < \frac{8}{5}\) and \(x > -\frac{4}{5}\). Together, they define the interval \(-\frac{4}{5} < x < \frac{8}{5}\).

Step 5: Express in Interval Notation

Convert the combined inequality to interval notation. The solution is \((-\frac{4}{5}, \frac{8}{5})\). This expresses all values of \(x\) that satisfy the original absolute value inequality.

Key Concepts

Interval Notation

Interval notation is a method of representing a set of numbers, specifically the solutions to inequalities in a compact form. It succinctly describes the range of values within an interval, using parentheses and brackets to indicate whether endpoints are included or excluded.

In the exercise above, the solution to the inequality \(|5x - 2| < 6\) lies between the values \(-\frac{4}{5}\) and \(\frac{8}{5}\). In interval notation, this is represented as \((-\frac{4}{5}, \frac{8}{5})\). This means:

• The parentheses indicate that the endpoints \(-\frac{4}{5}\) and \(\frac{8}{5}\) are not included in the interval.
• A comma separates the lower bound and the upper bound of the interval.

Understanding interval notation is crucial for properly communicating solutions to inequalities. It provides a clear visual cue for which values are part of the solution set and easily distinguishes between inclusive and exclusive boundaries.

Solving Inequalities

Solving inequalities involves finding the set of all possible values of a variable that make an inequality true. The process is similar to solving equations but requires careful attention to the inequality signs.

To solve an inequality:

• Perform operations similar to those used in solving equations (like adding, subtracting, multiplying, and dividing).
• When multiplying or dividing by a negative number, reverse the direction of the inequality sign.

For our inequality \(|5x - 2| < 6\), it was split into two separate inequalities: \(5x - 2 < 6\) and \(5x - 2 > -6\). By solving these separately, we found that \(x < \frac{8}{5}\) and \(x > -\frac{4}{5}\).

Once you have the solutions for both inequalities, you combine them to find the overall solution for the absolute inequality. This final step often involves understanding what values satisfy both conditions simultaneously.

Absolute Value Functions

Absolute value functions express the distance of a number from zero, without regard to direction. Mathematically, the absolute value of a number is defined as:

• If the number is positive or zero, its absolute value is the same number.
• If the number is negative, its absolute value is the number without its negative sign.

For example, the absolute value of both -3 and 3 is 3. In equations, the absolute value function is represented as \(|x|\).

In the given exercise, the inequality \(|5x - 2| < 6\) means that the expression \(5x - 2\) lies within 6 units of distance from zero. Solving absolute inequalities typically requires splitting the expression inside the absolute value into two separate inequalities.

• One for the positive case: \(5x - 2 < 6\).
• One for the negative case: \(5x - 2 > -6\).

This approach allows for capturing all possible values that satisfy the inequality. It's a fundamental concept in understanding how to manipulate and solve absolute value problems effectively.